Seven characterizations of non-meager $\mathsf {P}$-filters

Kenneth Kunen ¹   ; Andrea Medini ²   ; Lyubomyr Zdomskyy ²

¹ Department of Mathematics University of Wisconsin 480 Lincoln Drive Madison, WI 53706, U.S.A.
² Kurt Gödel Research Center for Mathematical Logic University of Vienna Währinger Straße 25 A-1090 Wien, Austria

Fundamenta Mathematicae, Tome 231 (2015) no. 2, pp. 189-208

Résumé

We give several topological/combinatorial conditions that, for a filter on $\omega $, are equivalent to being a non-meager $\mathsf {P}$-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager $\mathsf {P}$-filter. Here, we identify a filter with a subspace of $2^\omega $ through characteristic functions. Along the way, we generalize to non-meager $\mathsf {P}$-filters a result of Miller (1984) about $\mathsf {P}$-points, and we employ and give a new proof of results of Marciszewski (1998). We also employ a theorem of Hernández-Gutiérrez and Hrušák (2013), and answer two questions that they posed. Our result also resolves several issues raised by Medini and Milovich (2012), and proves false one “theorem” of theirs. Furthermore, we show that the statement “Every non-meager filter contains a non-meager $\mathsf {P}$-subfilter” is independent of $\mathsf {ZFC}$ (more precisely, it is a consequence of $\mathfrak {u}\mathfrak {g}$ and its negation is a consequence of $\Diamond $). It follows from results of Hrušák and van Mill (2014) that, under $\mathfrak {u}\mathfrak {g}$, a filter has less than $\mathfrak {c}$ types of countable dense subsets if and only if it is a non-meager $\mathsf {P}$-filter. In particular, under $\mathfrak {u}\mathfrak {g}$, there exists an ultrafilter with $\mathfrak {c}$ types of countable dense subsets. We also show that such an ultrafilter exists under $\mathsf {MA(countable)}$.

Détail

DOI : 10.4064/fm231-2-5

Keywords: several topological combinatorial conditions filter nbsp omega equivalent being non meager mathsf filter particular filter countable dense homogeneous only non meager mathsf filter here identify filter subspace omega through characteristic functions along generalize non meager mathsf filters result miller about mathsf points employ proof results marciszewski employ theorem hern ndez guti rrez hru answer questions posed result resolves several issues raised medini milovich proves false theorem theirs furthermore statement every non meager filter contains non meager mathsf subfilter independent mathsf zfc precisely consequence mathfrak mathfrak its negation consequence diamond follows results hru van mill nbsp under mathfrak mathfrak filter has mathfrak types countable dense subsets only non meager mathsf filter particular under mathfrak mathfrak there exists ultrafilter mathfrak types countable dense subsets ultrafilter exists under mathsf countable